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Contents of partitions and the combinatorics of permutation factorizations in genus 0


Authors: S. R. Carrell and I. P. Goulden
Journal: Trans. Amer. Math. Soc. 370 (2018), 5051-5080
MSC (2010): Primary 05A15; Secondary 05A05, 05E05, 14E20
DOI: https://doi.org/10.1090/tran/7143
Published electronically: February 21, 2018
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Abstract: The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $ f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $ \mathcal {U}$ and $ \mathcal {D}$ operators, whose action on the Schur symmetric function $ s_{\lambda }$ can be simply expressed in terms of powers of the contents of the cells in $ \lambda $. Among our results, we construct the $ \mathcal {U}$ and $ \mathcal {D}$ operators explicitly as partial differential operators in the underlying power sum symmetric functions. We also give a combinatorial interpretation for the content series in terms of the Jucys-Murphy elements in the group algebra of the symmetric group. This leads to an interpretation for the content series as a generating series for branched covers of the sphere by a Riemann surface of arbitrary genus $ g$. As particular cases, by suitable choice of the underlying series $ f$, the content series specializes to the generating series for three known classes of branched covers: Hurwitz numbers, monotone Hurwitz numbers, and $ m$-hypermap numbers of Bousquet-Mélou and Schaeffer. We apply our pde to give new and uniform proofs of the explicit formulas for these three classes of numbers in genus 0.


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  • [BMS00] Mireille Bousquet-Mélou and Gilles Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. 24 (2000), no. 4, 337-368. MR 1761777, https://doi.org/10.1006/aama.1999.0673
  • [C11] S. R. Carrell, Diagonal solutions to the 2-Toda hierarchy, Math. Res. Lett. 22 (2015), no. 2, 439-465. MR 3342241, https://doi.org/10.4310/MRL.2015.v22.n2.a6
  • [CG10] S. R. Carrell and I. P. Goulden, Symmetric functions, codes of partitions and the KP hierarchy, J. Algebraic Combin. 32 (2010), no. 2, 211-226. MR 2661415, https://doi.org/10.1007/s10801-009-0211-2
  • [DG89] P. Diaconis and C. Greene, Applications of Murphy's elements, Tech. Report 335, Dept. Stat., Stanford Univ., 1989.
  • [ELSV01] Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), no. 2, 297-327. MR 1864018, https://doi.org/10.1007/s002220100164
  • [Fa16] W. Fang, Enumerative and bijective aspects of combinatorial maps: generalization, unification and application, PhD thesis, Université Paris Diderot, 2016.
  • [Fé12] Valentin Féray, On complete functions in Jucys-Murphy elements, Ann. Comb. 16 (2012), no. 4, 677-707. MR 3000438, https://doi.org/10.1007/s00026-012-0153-6
  • [G94] I. P. Goulden, A differential operator for symmetric functions and the combinatorics of multiplying transpositions, Trans. Amer. Math. Soc. 344 (1994), no. 1, 421-440. MR 1249468, https://doi.org/10.2307/2154724
  • [GGN13a] I. P. Goulden, Mathieu Guay-Paquet, and Jonathan Novak, Monotone Hurwitz numbers in genus zero, Canad. J. Math. 65 (2013), no. 5, 1020-1042. MR 3095005, https://doi.org/10.4153/CJM-2012-038-0
  • [GGN13b] I. P. Goulden, Mathieu Guay-Paquet, and Jonathan Novak, Polynomiality of monotone Hurwitz numbers in higher genera, Adv. Math. 238 (2013), 1-23. MR 3033628, https://doi.org/10.1016/j.aim.2013.01.012
  • [GGN14] I. P. Goulden, Mathieu Guay-Paquet, and Jonathan Novak, Monotone Hurwitz numbers and the HCIZ integral, Ann. Math. Blaise Pascal 21 (2014), no. 1, 71-89 (English, with English and French summaries). MR 3248222
  • [GJ83] I. P. Goulden, D. M. Jackson, and Combinatorial enumeration, with a foreword by Gian-Carlo Rota, Wiley-Interscience Series in Discrete Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 702512
  • [GJ97] I. P. Goulden and D. M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), no. 1, 51-60. MR 1396978, https://doi.org/10.1090/S0002-9939-97-03880-X
  • [GJ00] Ian P. Goulden and David M. Jackson, Transitive factorizations in the symmetric group, and combinatorial aspects of singularity theory, European J. Combin. 21 (2000), no. 8, 1001-1016. MR 1797682, https://doi.org/10.1006/eujc.2000.0409
  • [GJ08] I. P. Goulden and D. M. Jackson, The KP hierarchy, branched covers, and triangulations, Adv. Math. 219 (2008), no. 3, 932-951. MR 2442057, https://doi.org/10.1016/j.aim.2008.06.013
  • [GJ09] I. P. Goulden and D. M. Jackson, Transitive powers of Young-Jucys-Murphy elements are central, J. Algebra 321 (2009), no. 7, 1826-1835. MR 2494750, https://doi.org/10.1016/j.jalgebra.2009.01.004
  • [GS06] I. P. Goulden and Luis G. Serrano, A simple recurrence for covers of the sphere with branch points of arbitrary ramification, Ann. Comb. 10 (2006), no. 4, 431-441. MR 2293649, https://doi.org/10.1007/s00026-006-0298-2
  • [H1891] A. Hurwitz, Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1-60 (German). MR 1510692, https://doi.org/10.1007/BF01199469
  • [J74] A.-A. A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Rep. Mathematical Phys. 5 (1974), no. 1, 107-112. MR 0419576
  • [KZ15] Maxim Kazarian and Peter Zograf, Virasoro constraints and topological recursion for Grothendieck's dessin counting, Lett. Math. Phys. 105 (2015), no. 8, 1057-1084. MR 3366120, https://doi.org/10.1007/s11005-015-0771-0
  • [K92] Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1-23. MR 1171758
  • [LT01] A. Lascoux and J.-Y. Thibon, Vertex operators and the class algebras of symmetric groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 156-177, 261 (English, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 121 (2004), no. 3, 2380-2392. MR 1879068, https://doi.org/10.1023/B:JOTH.0000024619.77778.3d
  • [La04] Michel Lassalle, Jack polynomials and some identities for partitions, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3455-3476. MR 2055741, https://doi.org/10.1090/S0002-9947-04-03500-7
  • [La13] Michel Lassalle, Class expansion of some symmetric functions in Jucys-Murphy elements, J. Algebra 394 (2013), 397-443. MR 3092727, https://doi.org/10.1016/j.jalgebra.2013.06.013
  • [M95] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., with contributions by A. Zelevinsky, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144
  • [Mu81] G. E. Murphy, A new construction of Young's seminormal representation of the symmetric groups, J. Algebra 69 (1981), no. 2, 287-297. MR 617079, https://doi.org/10.1016/0021-8693(81)90205-2
  • [Ok96] Andrei Okounkov, Young basis, Wick formula, and higher Capelli identities, Internat. Math. Res. Notices 17 (1996), 817-839. MR 1420550, https://doi.org/10.1155/S1073792896000505
  • [Ok00] Andrei Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), no. 4, 447-453. MR 1783622, https://doi.org/10.4310/MRL.2000.v7.n4.a10
  • [Or02] A. Yu. Orlov, Tau functions and matrix integrals, arXiv:math-ph 021001v3, 2002.
  • [W91] Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243-310. MR 1144529
  • [Z81] Andrey V. Zelevinsky, Representations of finite classical groups, A Hopf algebra approach, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. MR 643482

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Additional Information

S. R. Carrell
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada
Email: srcarrel@uwaterloo.ca, s.r.carrell@gmail.com

I. P. Goulden
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Ontario N2L 3G1, Canada
Email: ipgoulde@uwaterloo.ca

DOI: https://doi.org/10.1090/tran/7143
Keywords: Generating functions, transitive permutation factorizations, symmetric functions, Jucys-Murphy elements, contents of partitions
Received by editor(s): August 5, 2016
Received by editor(s) in revised form: November 11, 2016
Published electronically: February 21, 2018
Additional Notes: The work of the second author was supported by an NSERC Discovery Grant.
Article copyright: © Copyright 2018 American Mathematical Society

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